ELSEVIER Linear Algebra and its Applications 277 (1998) 313-336

LINEAR ALGEBRA AND ITS

APPLICATIONS

Time-varying terms of Ben

discrete Riccati equation in Artzi-Gohberg dichotomy

Vlad I o n e s c u 1

3 Emile Zola, 71272 Bucharest, Romania

Received 17 May 1996; accepted 6 October 1997

Submitted by M. Neumann

Abstract

Necessary and sufficient conditions for the existence of the stabilizing solution to the time-varying discrete Riccati equation are derived in terms of the so-called Ben Artzi Gohberg dichotomy. It is shown that the problem of such existence conditions is strong- ly related to that of unique solvability, in vector-valued f-spaces, of a system of singular difference equations which, in this case, is termed the extended Hamiltonian system (EHS). Connections between the existence of the stabilizing solution to the Riccati equa- tion and the bounded invertibility of an appropriate/Z-operator are pointed out via the notion of disconjugacy. © 1998 Elsevier Science Inc. All rights reserved.

K~:vwords: Time-varying systems; Riccati equation; Dichotomy; Hamiltonian systems; /2-spaces and f-operators

1. Introduction

In the last decade the topics on the Riccati equat ion theory have been contin- uously enlarged both for the t ime-invariant and t ime-variant situations. The present paper extends, to the t ime-varying case, the results obtained in [1-5] for the t ime-invariant case. It worth-while to be mentioned that the subsequent devel- opment may be also seen as a "matr ix pencil" alternative o f the t ime-variant Ric- cati theory developed in [6-9]. The main tool which will be intensively used is the

I E-mail: ionescu@karla.indinf.pub.ro.

0024-3795/98/$19.00 © 1998 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 7 ) 1 0 0 4 2 - 8

314 V. lonescu / Linear Algebra and its Applications 277 (1998) 313 336

dichotomy theory, developed by Ben Artzi and Gohberg in [10-12] and Ben Artzi et al. in [13]. Among the main objectives of the above cited theory is that of unique solvability, in vector-valued/2-spaces, of a system of singular difference equations. In our case such a system is the so called extendedHamil tonian system (EHS) and, it is proved that if the EHS admits a dichotomy then its rank equals always the dimension of the state-space to which the Riccati equation is related• This result combined with the property of disconjugacy (see for instance Coppel [14]) provide necessary and sufficient conditions for the existence of the stabiliz- ing solution to the time-varying discrete Riccati equation (TVDRE).

In the sequel the following notations will be used• By Z, ~(C), T, ~"(C n) and 0~ "×m (C "×m) we denote the ring of integers, the field of real (complex) numbers, the unit circle, the real (complex) n-dimensional Euclidian space and the set of n × m matrices with real (complex) entries• I f M E 0~ n×m then M T stands for its transpose and M -~- abreviates (M-I) T. By I, we denote the n × n unitary ma- trix. The spectral radius of a linear bounded operator T on a Hilbert space will be denoted by p(T) and T* will stand for the adjoint of T.

L e t 12'm be the Hilbert space of norm-square doubly infinite Cm-valued se-

quences u = (Uk)k~Z, i.e. uk 6 C m and [[b/]] 2 : = k ~ - ~]]U~H ~ < ~ is the l 2-

norm; here ]] ]] stands for the usual Euclidian norm• I f u E /2.m then we shall adopt for u the doubly infinite column representation u = col(uk)k~_~(Uk Ecm) . I f B is any linear bounded operator from 12'm into 12•" then, as usually it proceeds, we shall identify it with the doubly infinite

o c

block matrix representation B = mat(B~j)~j . . . . . B~ E C "×m. Thus the action of B, i.e. l 2̀ m ~ u ~ x = Bu E /2,n is described by the simple matrix multiplica-

1 ~ ~ 1 tion rule co (xi)~=_~ -- mat(Bi/)/j= ~co (u i) j . . . . or explicitly

x I

Xl

z

B-1,-1 B I.o

Bo. 1 [ - ~

Bi ,o

Bo, 1

Bj.~ ". ul |

• I

(1.1)

Let us introduce now on the Banach space of linear bounded operators from 12'm into 12n the pre and post action of the shift operator a and its inverse a - ~, in accordance with the following definition•

Definition 1.1. Let B be a linear bounded operator from 12'm into 12'n with the matrix representation B = mat(Bij)~ . . . . Bij E C n×m. Then: 1. ~B := mat(Bi+l,j+j)i,j=_~;

~ c

2. a IB := mat(Bi - l j - j ) i j=_~;

14 lonescu / Linear Algebra and its Applications 277 (1998) 313~36 315

o c

3. Ba := mat(Bi , j - l ) i j= ~,; 4. Ba -l := mat(Bi , /+l)3_ ~.

Convention: In an ope ra to r m o n o m i a l only the opera tors o f type Ba or Ba will be parenthesised. Fo r instance ( m ) ( a N ) ( P a ) ( a Q ) = M a N ( P a ) a Q .

O f ma jo r impor tance for our fur ther deve lopment are those opera tors that have doubly infinite block diagonal matr ix representat ion, that is,

o c

B = mat(f~jB~)g,/_ ~ = diag(B~)~ ~o, B~ E C . . . . and 6gj are the Kronecke r indi- ces. Clearly B is a linear bounded ope ra to r f rom 12'm into l 2,n if and only if the matr ix sequence B = (B~)~z is bounded, i.e. there exists co/> 0 such that ]]B~]I ~< co Vi E 7/; here 1111 stands for the matr ix n o r m induced by the Euclidian norm. Not ice that the above ment ioned opera tors act as multiplication opera-

o o tots. Indeed, Bu = diag(Bi)~=_~col(ui)~_~ = col(B~ui)i=_~ or, in the equivalent sequence representat ion, Bu = (Biui)gcz. As easily can be remarked the multipli- cat ion opera to r s can be natural ly identified with (bounded) matr ix sequences. This fact will be fully exploited further.

Proposition 1.2. I f B = mat(6i jBi) i~=_~ = diag(B/)i ~ - ~., (bounded) block diagonal operator f r o m l 2"m into l 2," then:

1. aB := mat(6~jB~+l)~:_~ = d ag(B~+l)~ . . . . . ; 2. a-~B := mat(6~/B,_~) ~ = diag(B, ~)~ ~;

i , j = - v , : - =

3. Ba = mat(6ij_lB~)~, , : ~; 4. Ba J = mat( f i , /+lBi)~ . . . . . .

Bi E C nxm, is a

Proof. The p r o o f follows trivially f rom Definit ion 1.1. []

i ~ c

Denote by Ii2 . . . . d ag(Im) ~ the identity opera to r o n l 2"m and introduce now the following definition.

Definition 1.3. The linear bounded ope ra to r I t 2 ~ : 12"m--+ l 2,m defines the act ion o f the bilateral shift cr on l 2m. Similarly Ii2,,,a-I : l 2`m --+ 12m defines the act ion of the inverse bilateral shift on 12"m.

According to Definit ion 1.3 we have the following Proposi t ion.

x 12.m Proposit ion 1.4. Le t u = col(uk)k= o~ E . Then." 1. The action o f Ii, , ,a on u, i.e. (Ii2.ma)U ~ au is given explicit ly by au = col(uk+l)k~_o~ or equivalently, i f u = (uk)k~z then (au)k = ua.+l. 2. The action o f I12.,,a Ion u, i.e. (It2.~a-1)u _~ a lu is given explicit ly by a - l u = col(uk-l)k~ ~ or equivalently, i f u = (Uk)kc~ then (a lu)k = uk-1.

Proof. (1) According to (3). in Proposi t ion 1.2 we have m" " I I2ma= at(6cj llm)i,j= oo" Hence ( I i 2 . ~ a ) u = m a t ( f i j _ l l m ) ~ ~ c o l ( u j ) ~

316 v. lonescu / Linear Algebra and its Applications 277 (1998) 313 336

3C ~C O0 = c o l ( ~ j = - ~ c g i , J l U j ) i = cc col (u i+l) i=_~. For (2) the p roo f runs similar- ly. []

We have the following Proposi t ion.

Proposition 1.5. L e t B : l 2,m ---+ 12'', A : 12,n ---+ 12,,, C : l 2'" ---+ 12'p be three linear

bounded operators. Then the f o l l o w i n g hold." 1. a (CB) = aCaB; 2. B(I:2,,,a) = Ba; 3. (Aa)B = A ~ B a , in particular(l ,~_,a)B =_ (a)B = aBa; 4. (Aa) I = a _ l A _ l a _ l prov ided A has a bounded inverse.

Proof. (1) Let D := CB. Then according to (1) in Definition 1.1 we have = )~ m aD mat(D/+x,./+, )~=_~ = m a t ( ~ i ~ _ ~ Ci+l,sBs,j+l i,j= ,~c = a t ( ~ = _ ~ Ci+l,

= m ~ • ~ r + 1Br+lj+l)iy= ~ at(Ci+j,,.+j)~ . . . . . ~ mat(B~+l,j+l)~.j= .~ = aCaB. (2) B(Ii2.°,a) = mat(Bi.~)i.~= ~mat(fi~./_,Im)s,~=_~. = m a t ( ~ . ~ _ ~ Bisfis,j 1)~= ~ = mat (Bi,/_ l) ~ ~,j=_~ = B a as follows from (3) in Definition 1.1. (3) (Aa)B ~ ~ ~.= = mat(Ai,~ 1)i ..... ~mat(Bsj)~,j= ~ = m a t ( ~ ~Ai,,_lB~j)i,j=_ ~ = mat ( g-'~' A B ~ ~ z_~,.=-~ i~ ,-+l,jji,j=_~ = mat(A~)i . . . . . . mat(B,.+l,j)~j=_~ = A ( a B a ) = AaBc: where for the last two equalities, (1) and (3) in Definition 1.1 have been combined. (4). Let the following equat ion (Aa)x = z be considered. By combining (2) of the present proposi t ion with (1) in Proposi t ion 1.4 we may write (AIl=,,a)x =A(Ii~.,,cr)x = A(~x) = z . Then ax = A - l z f rom which x = (It2°a - I ) A - l z = ( a - I A l a - t ) z as follows f rom the appropr ia te interpretat- ion of (3) o f the present proposi t ion, i.e. by replacing in (3) ~ by a -1. Hence (Aa) -l = o--1mo --1. []

Proposition 1.6. L e t A = diag(Ak)~=_~c, Ak E C n×n, be bounded. Then f o r i >~ 1 1. (Act) i = mat(:ikj_iA~Ak+,... Ak+i-j)k~. . . . .

A 2. ( A a - t ) i = mat(6~j+,AkAk 1 . . . k-i+l)k,j=_:~.

Proof. Equalities (1) and (2) follows trivially by iterating equalities (3) and (4) in Proposi t ion 1.2, respectively. []

In the next sections italic capital letters will be used exclusively to denote block diagonal operators or, equivalently, (bounded) matr ix sequences, More- over Definition 1.1 and Proposi t ion 1.5 will be frequently used.

The paper has been planed as follows. Section 2 deals with the main aspects concerning the d ichotomy of a system of singular difference equation. The ba- sic definitions and results given in [13,12] are slightly modified in order to be adapted to the context of the present topics. Section 3 introduces the not ion o f EHS and dichotomic EHS is stressed out as well. Section 4 contains the main result on the stabilizing solution to the T V D R E .

I~ lonescu / Linear Algebra and its Applications 277 (1998) 313 336 317

2. Exponential dichotomy

Let M = (Mk)k~> Mk E ~r~. Introduce

f M i i "" Mi, i > j ,

SM,/ := / ZMi i = j , (2.1) /

"'" M / - 1 , i < j

gi , j E 7/and call S~ the evolution operator of M.

Definition 2.1. Let M be as above. We shall say that M defines an exponentially stable (ES) (anticausal exponentially stable (AES)) evolution if there exist Po /> 1 and 0 < q < 1 such that ]]Sff]] <~poq i-i, gi >~j (llSffl] <~poq j-i, Vj >~ i).

Let M be introduced above. Then for Ma J and Ma introduced through Definition 1.1 in conjunction with Proposition 1.2, the following results hold (see [15]).

Proposition 2.2. (1) I f M defines an ES evolution then M is bounded, Ma 1 is a linear bounded operator from l 2,r into itself and p(Ma -1 ) < 1. (2) I f M defines an AES evolution then M is bounded, Ma is a linear bounded operator J?om l 2r into itself and p(Ma) < 1.

Proof. In order to have a selfcontained treatment we reproduce here, with our terminology, the proof given in [15].

(1) According to (2) of Proposition 1.6 one gets .

= S~+l,k_i+lx~_,. (2.2)

Using (2.2) we have the evaluation

II ')'xll 2, o = IIS;+, ,+,xk-,ll- k = - o c

2 2i 2 = P~q Itxll;.

k = 9c

Thus we deduce that

II(M~-')e/F ~< Poq i. (2.3)

Using now (2.3) one obtains • 1/~ p(M~r -1) = !imll(Ma-I)illl/i<~ !imP0 q = q < 1.

(2) The proof runs similarly as above but in this case we start with formula (see (1) of Proposition 1.6)

318 E lonescu I Linear Algebra and its Applications 277 (1998) 313-336

((M~)'x)~ = S~.k+~x~+,. []

Corollary 2.3. (1) I f M defines an ES evolution then Ii2 . . . . M a 1 is boundedly invertible on l 2"~ and

(I,2r - M o - l ) -1 = Z ( M o - 1)i. (2.4) i>~0

(2) I f M defines an A E S evolution then 11:,- - Ma is boundedly invertible on l 2'~ and

(I,2.r -- MO') -1 = Z ( M f f ) i. (2.5) i>~0

Let M = (M~)kez, N = (Nk)kcz, with M~. and Nk E W ×', be given. Assume M and N bounded and consider the system of difference equations

(Ma - N)w = z, (2.6)

where z = (zk)keZ, W = (Wk)k~Z with zk and wk E C r. Since M and N are bounded M a - N is a linear bounded operator on 12,'. We shall say that (2.6) in uniquely solvable in l 2"~ if for arbitrary z E l 2" there exists unique w E l 2,r for which (2.6) is fulfilled. Clearly this is equivalent to the existence of a bounded inverse of M a - N, i.e. w = ( M a - N) lz, Vz E l 2".

The problem of bounded invertibility of M a - N is now briefly examined in terms of Ben Ar tz i -Gohberg theory. For the beginning let us introduce the fol- lowing definition.

Definition 2.4. We shall say that Ma - N: 12," ~ l 2.r defines an exponentially dichotomic (ED) evolution of rank p(0 ~<p~< r) if there exist four matrix sequences U = (Uk)kc~, Z = (Zk)k~z, S = (Sk)k~Z, and T = (Tk)~czwith Uk and Zk E W×~,Sk E EP×P, Tk E ~(~ pl×(r-p) are bounded, U 1 and Z -l such that U and Z are well defined and bounded, S defines an ES evolution, T defines an AES evolution and

U ( M a - N ) Z = [ I '2"o- S Ta -O ,, (2.7)

Remark 2.5. From (2.7) one gets easily

( M a _ N ) _ , = Z [ ( 1 , 2 , , a - S ) ' 0 ]U, 0 (Ta - It2, ,,)-I

where

(2.8)

and

V. lonescu I Linear Algebra and its Applications 277 (1998) 313-336 319

(1124,~-S) - 1 = ((I t2. , , -Sa- ')a) ' = a - ' ( I p _ , , - S a ' ) - la l

= Z a - l ( S a - 1 ) i ~ -' (2.9) i~>O

(Ta - Iz-',-,,)-' = -(I,2.,-,, - Ta) ' = - Z ( T c r ) ' (2.10) i~>0

as directly follows f rom (4) of Proposi t ion 1.5 and Corol lary 2.3. Thus we may conclude from (2.8) that if M a - N defines an ED evolution then it is bound- edly invertible on 12,~.

It is interesting to note addit ionally that (2.9) hides the variat ion o f con- stants formula for ax = Sx + v (where S defines an ES evolution). Indeed using (2) in Proposi t ion 1.6 combined with (2) and (4) in Definition 1.1 one gets

x, = ((I,~_..a- S) iv), = cr '(Scr l ) ia- l )v \ ~=o /

= mat(6~.-14+i&-lSk 2 . . . & i)k.)= ~col(vi))~ ~ / t

CO k - I . j+i l 2 • . .

/ k = - . ~ / I

= col jSk 2...Sk-iV*-ilk=_~ /

3c 3~

: Z S , - I S I - 2 . . . S I i v, i - , : Z S ~ . , i V l i , i=0 i=0

/ - I

= Z s,% v,

Now we need the following proposit ion.

Proposition 2.6. I. The following two assertions are equivalent." 1. Ma - N defines an ED evolution of rank p," 2. There exist four matrix sequences V = (Vk)k~z, W = (Wk)k~,,S ~r(Sk)k~z, and T = (Tk)kcz with Vk E ~rxp, mk E ~rx(r-p), Sk E "~pxp, Tk E ~ l p) I -p) such that the following three conditions all hold." (a) S defines an ES evolution and T defines an A E S evolution," (b) IV W] and [MaV NW] are bounded with IV W] -I and [MaV NW] -l well defined and bounded." (c) the following equations hold

320 Id Ionescu / Linear Algebra and its Applications 277 (1998) 313 336

NV = MAPS, (2.11)

M a W = NWI'. (2.12)

II. I f I (2) holds' then

2@lf (M~)(Z~ta- N) ' d,~ T

/ 13t

' / o o] 2~i ( 2 M a - N ) - ~ M a d ) ' = [ V W] IV W] -~, (2.14) T

that is, the left-hand sides o f Eqs. (2.13), and (2.14) express, in terms of the original data, the projections onto M~V along NW and onto V along W, respec- tively. Thus if Ma - N admits a dichotomy then its elements are well defined in terms of the original data M and N.III. Assume that 1(2) holds. For any w - (wi)i~z and k E • let w ~ := (wi)~>~k and kW := (Wi)i~k 1" Here w, E C ~. Then."

ImVk = { ~ E C " : 3 w k with wk = ~ such that (ma - N)w k = O and wi ~ 0 (i ~ oc)},

(2.15)

I m W k = { ~ E C " : 3kw with wk : = ~ s u c h that ( M a - N)( kw) =O and wi ~ O(i ~ -oc)}

for all k E 77. Hence Im Vk and Im W~ are uniquely defined for all k E 7/.

(2.16)

Proof. I. (2) ~ (1). Let U := [MaV NW] -I, Z := IV W]. Then using (2.11), (2.12) and recalling that (Ma)V ~ Ma V~r and (Ma)W = Ma W~z (see Section 1) one obtains

U ( M a - N ) Z = [MaV NW] ' ( M a - N ) I V W]

= [MaV N W ] - ' [ M c r V a - NV M a W a - NW]

= [MaV N W ] - I [ M a V a - MaVS N W T a - NW]

= [McrV NW] '[MaV(I,2. , ,a-S) NW(-1,,~.,..-,,+Ta]

Id Ioneseu / Linear Algebra and its Applications 277 (1998) 313 336 321

Proof. I. (1) ~ (2). Let U 1 and Z be par t i t ioned as U l = [ L Z = [ V W 1, respectively with Lk and I~. E Er×p. Then (2.7) yields

o r

R ] and

(Mcr - N)Z = U-1[ II2'P; - S T6 -O ,,

( M a - N ) [ V W ] = [ L R][II2P; - S Ta -O

=[La-LS Rr~-R] f rom which by identifying the lef tmost and the r ightmost terms entry by entry and by taking into account Proposi t ion 1.5 one gets

( M a - N ) V M a V 6 - NV = L a - LS and ( M a - N ) W

= M a W 6 - N W = RYe - R. (2.18)

Using now the appropr ia te matr ix representat ion in Proposi t ion 1.2 (2.18) yields

M a V a = La, (2.19)

NV = LS, (2.20)

M 6 W a = RTa, (2.21)

NW = R. (2.22)

F r o m (2.19) and (2.22) we deduce L = M~rV, R = NW, i.e. U J = [MaV NW] and (2(b)) follows. Fur ther (2.20) and (2.21) yield N V = M a V S and M a W = NWF, respectively and (2(c)) follows as well.

Proof II . Since I(2) holds M a - N defines an E D evolut ion of rank p and the bounded invertibility o f M 6 - N follows f rom R e m a r k 2.5 (see (2.8)). For each )o E T in t roduced P(2) by (P(2)w)k = 2kwk, Vw = (wk)kc~, wk C C r. Clearly P()o) is a uni tary opera to r on l 2r and

2M6 - N = P*(2)(M6 - N)P(2) V). C T (2.23)

because of,~k2 k+t = e-ik°ei(~+lt° = e i° = 2 (2 is the conjugate o f 2). As Ma - N is invertible on l 2,r, (2.23) shows that 2 M ~ - N is also invertible on l 2'' for all 2 E T. Let us check (2.13). Using (2.8), (2.9) and (2.10), with a replaced by 26 in th lef t-hand sides of (2.9) and (2.10), one gets

322 V. Ioneseu / Linear Algebra and its Applications 277 (1998) 313 336

( ~ a _ N ) , [V W]I(21F_.pa-S)-' 0 ] 0 ()oTa - I~:,,-,,)-1

[MaV NW]-'

[ Z 2-'-'o-'(s~-')'o-' o ] : [ V W] i~>O

o - ~)~(r~)' i>~O A

[MaV NW] '. (2.24)

We have also

(Ma)[V W] = [MaVa MaWa] = [MaVa NWTa]

as follows from (2.12). Combining (2.24) and (2.25) one gets

Ma()dl4a - N) J

(2.25)

=[MaV NW][ I1"'a O] [ y'~2-'-'a-'(Sa ')ia ' 0 o w ~>~o o -Z)/(w) ~

i~>0

[MaY NW] '

=[MaV NW 1 -2-111~,,, + ~ 2-i-,(Sa 1)i 0

i ~ l

o - Z ;~i(T~)'+' i>~O

[MaV NW] ', (2.26)

where for the upper left corner of the middle matrix factor in the rightmost term of (2.26) the following computation has been performed in accordance with (3). of Proposition 1.fi: (/,2[,a) ~i >-o )'-i-la '(Sal-')ia ' = Z i > ~ 0 ) . i I(so._l)i /~ l i ,_~p+~i>,2 ' (Sa-). Since" ( A M a - N ) - is lo- cally analytic for every 2 ¢ C with ]2l > p and f3 < 1, the power series that ap- pear in (2.26) are absolutely and uniformly convergent on T. Hence they can be integrated term by term. Further, since

1 f2, d)~= {0, k :~ -1 , 2rti 1, k = -1 ,

T

v. lonescu I Linear Algebra and its Applieations 277 (1998) 313~36 323

formula (2.13) follows easily from (2.26). A similar computation is performed for checking (2.14).

Proof III. Let ~ ¢ I m Vk. Then

[Vk W~] t ~ = [ ¢ , ] } P . (2.27) 0 }r - p

Define the sequence w* by

with the sequence w~l defined by

aw]= Sw], wa,k = 3,. (2.29)

Then (2.17), (2.27), (2.28) and (2.29) imply

w~ = ~ (2.30)

and

[McrV XW]- ' (Mcr- N)w*

=[M~V NW]- I (Mty -N)[V WI[V m]-lw k

that is,

(Ma - N)w k = 0. (2.31)

Since S defines an ES evolution it follows from (2.29) that w~.i -~ 0 as i -~ w. Hence w, ~ 0 (i ~ oc) as follows from (2.28). This conclusion combined with (2.30) and (2.31) shows that Im V~ is contained in the right-hand side of (2.15).

Conversely, let ~ be in the right-hand side of (2.15) and let w ~ be such that wk = ~, (Mcr - N)w k = 0 and wi ~ 0 as i ~ oo. Let also

w~J = IV W]-lw k (w,.i ¢ CP, w2,i E C r P,i ~> k). (2.32)

Then with (2.17) and (2.32) we get

O=[MaV NW] ' ( M ~ - X ) w ~

=[MtrV N W ] - ' ( M a - N ) [ V W][V W] 'w k

324 V. lonescu I Linear Algebra and its Applications 277 (1998) 313-336

=?o" ° l[ l TO; - 112, p

or equivalently

(TW~ = SW], W'~lk = ~1, (2.33)

w~ = TawS, w~, k = ~2, (2.34)

where

P { I ~ I I =[Vk Wk] '~. (2.35) r - p{

According to (2.1), (2.34) yields

~2 = S[~w2,i Vi >~ k, (2.36)

where here S r is the evolution operator associated with T. Since wi ~ 0 as i--~ oc it follows that w k is bounded. Hence w~2 is bounded as well (see (2.32)) and Ilwe,ill~<c0 Vi>>-k for an appropriate co. On the other hand ]]S[i]t<,pq i-k i ) k , for appropriate p>~ 1 and 0 < q < 1, as follows from the fact that T defines an AES evolution. Therefore (2.36) yields

ll~-2ll ~< Ils£]l Ilw2.;l[ <~pcoq '-k (i >~ k). (2.37)

By taking i --+ ~ , (2.37) yields 42 = 0 and (2.35) shows that ¢ c Im Vk. Thus the right-hand side of (2.15) is contained in Im Vk and the conclusion follows. The proof of (2.16) is entirely symmetric. []

Remark 2.7. II of Proposition 2.6 concordes with the original definition of dichotomy given by Ben Artzi and Gohberg in [10,12]. The proof III of Proposition 2.6 is fully inspired from [12], Proposition 4.1.

As has been pointed out in Remark 2.5 the property of dichotomy is a sufi- cient condition for the bounded invertibility of Mo- - N. The converse is also true. To be more specific let us state the fundamental result due to Ben Artzi and Gohberg (see [12], Theorem 2.1)

Theorem 2.8. L e t M = (Mk)kcz , N = (Nk)kcz be bounded (Mk, Nk E ~r×~). Then the fo l lowing two assertions are equivalent."

1. M a - N has a bounded inverse on l 2'', 2. Mcr - N defines an E D evolution.

Remark 2.9. In fact the implication (2) ~ (1) in Theorem 2.8 has been proved in Remark 2.5. The heavy implication is (1) ~ (2) and the proof of it is

Id lonescu / Linear Algebra and its Applications 277 (1998) 313~36 325

essentially based on the spectral decomposition theorem for operator pencils which appears in Gohberg and Kaashoek [16].

Finally let us emphasize the fact that Definition 2.4 generalizes the classical concept of dichotomy specific for the time-invariant case. Recall that a real square matrix is said dichotomic (in the discrete case) if it has no eigenvalues on T.

Thus i f M a - N reduces to l a - N where N E W ×r then (2.12) becomes

W - NWF. (2.38)

As W is of full column rank T must be nonsingular and (2.38) provides

WT ~ = NW. (2.39)

With (2.39) in (2.17) one gets

IV WT-']-I(Ia-N)[V W]= I laoS TaO ]_i

or equivalently

[V W] ~(Ia-N)[V W ] = [I~roS 0 ] Ia - T -l "

Hence

[Z E; o] V-' ' (2.40)

where S and T i have their eigenvalues inside and outside the closed unit disk. respectively. Thus (2.40) reveals the dichotomy of the matrix N.

3. Extended Hamiltonian systems

For a systematical treatment two definitions are in order as follows.

Definition 3.1. A triplet Z=(A,B;P), where A=(Ak)kc~, B : ( B k ) ~ c z , P : (Pk)~-~z are bounded matrix sequences with Ak E ~n×n, Bk C R n×m,

[ Qk Lk ] = p [ E ~(n+m)×(n+m, = Rk

will be called a Popov triplet.

The extensive notation Z = (A, B; Q, L, R) will be also used.

326 V. Ionescu / Linear Algebra and its Applications 277 (1998) 313-336

Definition 3.2. Let Z be a Popov triplet. The system of singular difference equations (2.6) where M and N receive the particular values

F '°i]o [i (3.1)

-B* O R

(Mk and Ark E ~(2n+m)×(2n+m)) is called the EHS associated with Y. In this case Mcr - N will be termed as the extended Hamiltonian operator (EHO).

Remark 3.3. The system (3.1) is originated by the classical linear quadratic problem (LQP) which consists in the following. To a given Popov triplet ~. = (A ,B;P) associate the controlled discrete system

~x = Ax + Bu, xk,~ = ~. E ~n

together with the quadratic cost criterion

1 y~ kYk, Y k = [ X r Uk] T

>~ ko

(k0 E 7/). With k0 and ¢ specified find u = (u,)~ >~ k0 for which J attains its min- imum. The Hamiltonian of the problem is

H(x, u) = 5 Ix +

from where the canonical system is easily derived. Thus one gets

ax = H~(x, or2, u) = Ax + Bu,

2 = H x ( x , a2, u) = Q x + A * a 2 + L u ,

0 = H,(x , a2, u) = L*x + B*c~2 + Ru,

where Hj., Hx and Hu are the gradients with respect to 2, x and u, respectively. Rearranging the above system one obtains

( m a - N ) w = 0

?- T with M and N given explicitly by (3.1) a n d w k = [x r 2 r u k] .

The main result of this section is expressed by the following theorem.

Theorem 3.4. I f the E H O M ~ - N defines an ED evolution o f rank p then p = n. In order to prove Theorem 3.4 we need first the following proposition.

l~ lonescu I Linear Algebra and its Applications 277 (1998) 313-336 327

Proposition 3.5. Assume that the E H O defies an ED evolution o f rank p and let the matrix sequences V and W, introduced in I (2 ) o f Proposition 2.6, be partitioned in accordance with M and N in (3.1), that is

V = V~ , W = W2 (3.2)

W3

(Vi,k, V2,k 6 ~"×P, ~,~ E ~m×p and Y_ ). Then

Vl*~ = ~*v~

and

WI,k , m2,k C ~n×(2n+m-p) m3,k E ~mx(2n+m-p)Vk

(3.3)

W~* W2 - W2* W~. (3.4)

Proof. We shall use Proposition 2.6. Thus, in order to prove (3.3) write (2.11) for M and N given by (3.1) and obtain explicitly

AVt +BV3 = ~V~S,

QV~ - ~ + L ~ = - A * a ~ S , (3.5)

L*V~ + RV3 = - B ' a V i S .

From each equation in (3.5) we obtain successively

S*a(~*VI)S = S*~VjAVj + S* a~*BV3,

- V;* VI = - VI* Q Vj - V3* L* V~ - S* a V2* A VI ,

0 = -V~*L~ - ~*RV3 - S*aV2*B~.

By summing the left and right-hand sides of the above three equations one gets

S*a(V,*VI)S- ~*V~ = -~), (3.6)

where

0 := VI*QVI + VI*LV3 + ~*L*V1 + ~*RV3 = Q*.

Since S defines an ES evolution, the Lyapunov equation S*aXS - S + 0 = 0 has a unique (global and bounded on Z) solution X which in addition is self- adjoint because of the selfadjointness of 0 (see [7], Theorem 1.5.2). Hence by uniqueness arguments X = V2* VI must fulfill (3.6) and (3.3) follows.

For (3.4) use now the explicit form of (2.12) and obtain

aW~ = A W ~ T + B W 3 T ,

-A*aW2 = Q W ~ T - W2T + LW3T, (3.7)

-B* aW2 = L*WjT + RW3T.

328 V. hmescu I Linear Algebra and its Appfications 277 (1998) 313~36

Usir~g the first equation in (3.7) one gets

a(WI*W2) = T*WI*A* a W 2 + T*W3*B* crW:

= T*W~*(-QW~ T + W2T - LW3T) + T*Wj( -L*Wt T - RW3T)

= T*(WI*W2)T- T*(W1*QWI + WI*LW3 + W~L*W~ + W~RW3)T,

(3.8) where in the right-hand side of the first equality in (3.8), A*aW2 and B * a ~ have been substituted by their expressions given by the second and third equation in (3.11), respectively.

Thus (3.8) provides finally

a(W~*W2) - T*(W~*W:)T = -Q, (3.9)

where

O = T*(WI*QWJ + WI*LW3 + W3*L*W1 + ~ * R N ) T = (2".

Since T defines an AES evolution, the Lyapunov equation aY - T*YT + Q = 0 in which Q = Q* has a unique (global and bounded on 7/) selfadjoint solution Y (see [7], Theorem 1.5.3). Hence by uniqueness arguments as above the validity of (3.4) follows.

Now we are ready for proof of the theorem.

Proof of Theorem 3.4. Since

i-! o]E M a V = -A* c~Vi c~V~

--B*

is one to one ([Mc~V NW] is bounded invertible) it follows that

is one to one. From (3.3) we get

As ~,k ]

[~.kJ is of full column rank (see (3.10)),

[- Vj,k J

(3.10)

(3.11)

V. hmescu / Linear Algebra and its Applications 277 (1998) 313~36 329

is of full column rank too. Hence (3.11) shows that

p<~ dim KerIVir~ ~rk] =2n--rankfVir~ ~.rk] = 2 n - - p .

Hence

p<~n. (3.12)

Since

I ] is one to one ([ V W] is bounded invertible) it follows that both

and

are of full column rank. Using (3.4) one gets

L ~,~ Hence, if x E Ker W3,k, (3.13) shows that

F k] _ ~ . k [ x c K e r f ~ . ~ r k ~ r ] .

~.h-J Hence

--WLk Kerm.k C K e r [ W f W2 r. w3r]. (3.14)

W3,k

Since the matrix in the left-hand side of the inclusion (3.14) is of full column rank, as already has been noticed, (3.14) yields

dim Ker W~k.k ~< dim Ker[ W, rl,k W2~k Wr3.k ] •

330 l~ Ionescu / Linear Algebra and its Applications 277 (1998) 313~36

Hence

2 n + m p - r a n k W ~ k ~ < 2 n + m - r a n k [ W , r - - . . 1 A

+ m - p ) = p

and we have that

2n + m - 2p <~ rank W3k ~<m

or equivalently

n<~p.

By comparing (3.1 2) and (3.1 5) we conclude that n = p and the proof ends.

wT (2n 3,kl = 2 n + m -

(3.15) []

4. The time-varying discrete Riccati equation

Let 22 = (A, B; Q, L, R) be a Popov triplet. The following system associated with Y~

A*aXA - X + Q + (L + A* aXB)F -- O.

B*aXA + L" + (R + B*aXB)F = 0, (4.1)

where X and F are the unknowns, is called the Lur'e O,stem. Any pair of bound- ed matrix sequences ( X , F ) , X = (Xk)kcz,F -- (Fk)k~z,Xk ¢ R "×', Fk E ~m×, for which X X* and (4.1) is fulfilled and, in addition, A + B F defines an ES evolution, is called a stabilizing solution to (4.1). If (X, F) and ()(, F) are two

^

stabilizing solutions to (4.1) then (see [7], Proposition 3.1.6)X - X . For 2 given above associate also the TVDRE

A*aXA - X - (L + A~aXB)(R + B*crXB)-" (B*aXA + L*) + Q = 0. (4.2)

Recall that any selfadjoint solution X(= X ~) to (4.2) for which R + B*aXB has a bounded inverse and for

F := - ( R + B*aXB) I(B*aXA + L*). (4.3)

A + BF defines an ES evolution, is called a stabilizing solution.

Remark 4.1. If (X,F) is any stabilizing solution to the Lur'e system (4.1) such that R + B*aXB has a bounded inverse, then X is the unique stabilizing solution to the TVDRE (4.2) and F is related to X by (4.3) and it is unique as well. Conversely, if X is the stabilizing solution to (4.2) then the pair (X,F), with F given by (4.3), is a stabilizing solution to (4.1).

Proposition 4.2. Let Y~ be a Popov triplet and assume that the associated Lur'e ~3'stem has a stabilizing solution (X, F). Then the E H O M a - N associated with

V. lonescu / Linear Algebra and its Applications 277 (1998) 313 336 331

5" is bounded invertible o n 12"2n+m ( / 'and only i f R + B* aXB is bounded invertible on l 2"m .

Proof. Let ,4 := A + B F which clearly defines an ES evolution. Now it is easy checkable that (4.1) is equivalent to

A* c s X A - X + Q + LF + F*L* + F*RF = O,L* + R F + B* c~X,4 = 0 .

(4.4)

Let G and H be introduced by

i oo l,!oo] G = ]A a X Ii:,, F* , H = Ii'-,, 0

L H*ax o 112 .... o It:,,

which clearly are both bounded invertible o n 122 ...... . Using (4.4) one obtains via (3) of Proposition 1.5

o _ -8 ] G ( M a - N ) H = [ O 112., - A*a 0 .

- B * a - ( R + B*aXB)

(4.5)

As both I/2°a - ,4 and Ii2,, - A ' a are bounded invertible o n l 2'' (see Corollary 2.3) it follows from the zero structure of the right-hand side of (4.5) that this term has a bounded inverse o n 12.2,,+m if and only ifR + B*aXB is bounded inv- ertible on l 2". Hence the conclusion follows directly from the equality (4.5). [~

Let us introduce now the following definition.

Definition 4.3. Let Z be a Popov triplet and let Mc~ - N be the associated EHO. Assume that M a - N defines an ED evolution (of rank n). We call the EHO disconjugate if V1 in (3.2) is bounded invertible on l 2''.

Two remarks are now in order as follows.

Remark 4.4. The definition is consistent as follows from Theorem 3.4 which implies that Vl,k is n x n, i.e. is square for all k.

Remark 4.5. The definition of disconjugacy is independent on the particular choice of V = (Vk)kcZ as trivially follows from the uniqueness of (Im F~)kcz (see III of Proposition 2.6).

The main result of this paper is as follows.

Theorem 4.6. Le t 5: be a Popov triplet. Then the associated T V D R E (4.2) has a

(unique) stabil izing solution ( f and only i f the E H O associated with Y is

disconjuga te.

332 V. loneseu / Linear Algebra and its Applications 277 (1998) 313-336

Proof. "If"'. If the EHO is disconjugate, then (3.5) holds and, in addition, Vl is bounded invertible. Let

X : = ~V 1 I (4.6)

and

F := ~V 1 '. (4.7)

Then (3.5) yields

A 4 - B F = S ,

Q - X + LF -A*c;XS, (4.8)

L* + RF -- -B* aXS,

for

:= o-VISVj I. (4.9)

But X - X* as directly follows from (3.3) and S defines an ES evolution since it is Lyapunov similar to S (see (4.9)). By substituting S given by the first equa- tion (4.8) in the next two ones one can easily conclude that (X, F), defined by (4.6), (4.7), is a stabilizing solution to the Lur'e system (4.1). Since the EHO defines an ED evolution it is bounded invertible o n 12̀ 2n+m as follows from The- orem 2.8. Hence by using Proposition 4.2, R + B*aXB is bounded invertible and the conclusion follows from Remark 4.1.

"Only if''. Let X be the stabilizing solution to the TVDRE and let F be de- fined by (4.3). Then according to Remark 4.1 (X, F) is a stabilizing solution to the Lur'e system (4.1) which easily can be rewritten in the form presented in (4.8). Then for

1~ = (4.10)

(4.8) yields

N ( / = M A P S (4.11)

with S defining an ES evolution. On the other hand, since X is the stabilizing solution to the TVDRE, R 4- B*aXB is bounded invertible. Hence, according to Proposition 4.2 the EHO is bounded invertible o n l 2'2n+m. Using now Theorem 2.8 it follows that the EHO defines an ED evolution (on rank n). To prove the disconjugacy of the EHO it suffices to show that

Im V = I m V, (4.12)

where V is a constitutive element of the dichotomy (see I(2) of Proposition 2.6). Indeed, by comparing (4.10) with V given in (3.2) the bounded invertibility of

V. loneseu / Linear Algebra and its Applications 277 (1998) 313-336 333

Vl follows f rom (4.12). Let us prove now (4.12). For, let k E 2e and let C Im 4 , i.e. there exists ~j E E" such that

~- Vk~l. (4.13)

Let ~ = (wl.~)i >/k and w k = (wj)~ ~> ~. be defined by

aw~l " = Sw~, w~~ - ~t (4.14)

and

w k = /)w4, . (4.15)

Then with (4.14) and (4.15), (4.11) yields

X I)w~t " = M a ( / S w ~ = M a ( l ) w ~ )

o r

N w k = M a w ~ .

Hence

( M a - N ) w k = 0. (4.16)

But w~., --, 0 as i ~ oc since S defines an ES evolution (see (4.14)). Hence

wi ---+ 0 as i -~ oc (4.17)

as follows f rom (4.15). By combining (4.13), (4.16) and (4.17) it follows that Im ~. C Im Vk. As both Im ~ and Im Vk share the same dimension n, the above inclusion implies Im ~ = Im Vk Vk E 2, and (4.12) follows. Thus the "only if ' ' part is proved and the p r o o f o f the theorem ends. []

5. The time-invariant case

In this section we shall briefly point out how the results presented in [1], for the t i m e - i n v a r i a n t case, can be easily recaptured.

Notice first that in the t ime-invariant case M and N in (2.6) are r × r con- stant matrices with real entries. Notice also that in this case the following (usu- al) convent ion is adopted: if N E R n×" and w = (Wk)k~Z, Wk E E" then Nw := (Nw~.)~ez . If w E l 2'r then we denote by ~, the (discrete) Fourier trans- form o f w, that is, ~(2) = ~ [w] = ~ k c z wk2 ~, ~ E T and ~' E 2 (T,C") - the Hilbert space o f norm-square integrable C " - valued function defined on T. Hence, for z E 12"r and the unknown w E l 2 ' (if it exists), (2.6) becomes, in terms of Fourier t ransform,

( ~ M - N ) ~ , = ~. ( 5 . 1 )

334 14 hmescu / Linear Algebra and its Applications 277 (1998) 313~36

Now it is easy to see that 2M - N: L2(T, C") ~ L2(T, C r) is bounded invertible if and only if the matrix pencil £ M - N is regular (det(2M - N) ¢ 0) and it has no generalized eigenvalues on T.

Recall (see for instance [17]) that the Kronecker canonical f o r m of any regu- lar matrix pencil £ M - N is

( ; .M N ) - ; w E - 1 '

where J E ~"*×"" is in real Jordan canonical form and E E R .. . . . . . . is a block di- agonal nilpotent matrix, each block consisting of units placed on the first upper diagonal and zeros in the rest. Here nf and n~ are the number of finite and in- finite generalized eigenvalues. ByA~ we denoted the strict equivalence relation i.e. ( 2 A - B ) , - ~ ( 2 A - B ) , A , B , A , B E R p×q, if there exist U ¢ R p×p and V ~ R 't×q, both nonsingular, such that ),~i - / ~ = U(2A - B ) V . If £ M - N has no generalized eigenvalues on T, then we may write

where JL and ~ have all the eigenvalues placed inside and outside the closed unit disk. Then (5.2) combined with (5.3) yield

( . ~ - N) ~ 2 T - I

with

S : = J I , T : = j l

and where both S and T defines ES evolutions, i.e. their eigenvalues are placed inside the closed unit disk. But (5.4) is exactly the time-invariant counterpart of (2.7) and, expresses in this case, the significance of Ben Artzi-Gohberg dichot- omy.

If ,;~M - N is the ex tended symplec t i c penci l (ESP), i.e.

- B f L r O

(see [1]) then, as [11, Proposition 3 asserts, the ESP is dichotomic if and only if S in (5.4) is n x n. But this result is exactly the time-invariant version of Theorem 3.4. Further, Theorem 4.6 corresponds to Theorem 1 in [1] and formally they have identic statements.

I~ loneseu / Linear Algebra and its Applieations 277 (1998) 313 336

6. Conclusions

335

Based o n the c o n c e p t o f d i c h o t o m y i n t r o d u c e d by Ben Ar tz i a n d G o h b e r g , a n a t u r a l gene ra l i z a t i on to the t i m e - v a r i a n t case o f the resul ts o b t a i n e d in [1], for the t i m e - i n v a r i a n t case, has been achieved. As it has been s h o w n in Sect ion 5,

the theory deve loped in the p resen t p a p e r co r r e sponds , in the t i m e - i n v a r i a n t case, to the case o f regular mat r ix pencils . However , the t i m e - v a r i a n t c o u n t e r -

pa r t o f the Ricca t i t heo ry deve loped in [3] for the case w h e n the ESP (5.6) is singular r ema ins a n open p r o b l e m which is n o w u n d e r inves t iga t ion .

Acknowledgements

The a u t h o r is very i n d e b t e d to the referee for his va luab l e sugges t ions con- c e r n ing the accuracy o f the p resen t paper .

References

[1] V. Ionescu, M. Weiss, On computing the stabilizing solution of the discrete-time Riccati equation, Linear Algebra Appl. 175 (1992) 229-238.

[2] V. Ionescu, M. Weiss, Continuous and discrete-time Riccati theory: A Popov function approach, Linear Algebra Appl. 193 (1993) 173 209.

[3] V. lonescu, C. Oara, Generalized-discrete time Riccati theory, SIAM J. Control Optim. 34 (2) (1996) 601 619.

[4] V. lonescu, Reverse discrete-time Riccati equation and extended Nehari problem, Linear Algebra Appl. 236 (1996) 59-94.

[5] C. Oara, Stabilizing solution to the reverse discrete-time Riccati equation, Linear Algebra Appl. 246(1 3) (1996) 113-130.

[6] A. Halanay, V. lonescu, Generalized discrete-time Popov-Yakubovich theory, Systems Control Lett. 20 (1993) 1 6.

[7] A. Halanay, V. lonescu, Time-varying discrete linear systems, Birkhauser, 1994. [8] A. Halanay, V. lonescu, Anticausal stabilizing solution to discrete reverse-time Riccati

equation, Computers and Mathematics with Applications, Special Issue on Difference Equation, vol. 28 (1 3), Pergamon Press, Oxford, 1994, pp. 115-125.

[9] V. Ionescu, Generalized time-varying Popov-Yakubovich theory: The discrete case, Proced- ings of the MTNS'93, vol II, Akademie, 1994, pp. 717-720,

[10] A. Ben Artzi, I. Gohberg, Inertia theorems for nonstationary discrete systems and dichotomy, Linear Algebra Appl. 120 (1989) 95 138.

[I 1] A. Ben Artzi, I. Gohberg, Nonstationary inertia theorems, dichotomy and Applications, Proc. Sympos. Pure Math. Part I 51 (1990) 85-95.

[12] A. Ben Artzi, 1. Gohberg, Band matrices and dichotomy, Operator Theory: Advances and Applications, vol. 50, Birkhauser, 1991, pp. 137-170.

[13] A. Ben Artzi, I. Gohberg, M.A. Kaashoek, Invertibility and dichotomy of singular difference equations, Operator Theory: Advances and Applications, vol. 48, Birkhauser, 1990, pp. 157- 184.

[14] W.A. Coppel, Disconjugacy, LNM, vol. 220, Springer, Berlin, 1971.

336 I~ loneseu / Linear Algebra and its Applications 277 (1998) 313 336

[15] J. Ball, I. Gohberg, M.A. Kaashoek~ Nevalinna-Pick interpolation for time-varying input- output maps: The discrete case, Operator Theory: Advances and Applications, vol. 56, Birkhauser, 1992, pp. 1-51.

[16] I. Gohberg, M.A. Kaashoek, Blok Toeplitz operators with rational symbols, Operator Theory: Advances and Applications, vol. 35, Birkhauser, 1988, pp. 385-440.

[17] F.R. Gantmaher, Matrix theory (Russian edition), Nauka, Moskow, 1966.